# chowClass -- Finds the (fundamental) class of a subscheme in the Chow ring of the ambient space

## Synopsis

• Usage:
chowClass(IX)
chowClass(IX,A)
• Inputs:
• IX, an ideal, an ideal in the multi-graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
• A, , the Chow ring of \PP^{n_1}x...x\PP^{n_m}. This ring can be built by applying makeChowRing to the coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
• Optional inputs:
• Strategy (missing documentation) => ..., default value multidegree, multidegree, using "prob" uses a probabilistic method which is sometimes faster on large examples
• Outputs:
• isMultHom, , the class [X] in A where X is the subscheme associated to IX

## Description

Given a subscheme X of \PP^{n_1}x...x\PP^{n_m} this method computes [X] in the Chow ring of \PP^{n_1}x...x\PP^{n_m}.

 i1 : R=makeProductRing({6}) o1 = R o1 : PolynomialRing i2 : x=gens(R) o2 = {a, b, c, d, e, f, g} o2 : List i3 : J=ideal(x_0*x_2-x_4*x_5) o3 = ideal(a*c - e*f) o3 : Ideal of R i4 : clX=chowClass(J,Strategy=>"prob") o4 = 2H 1 ZZ[H ] 1 o4 : ------ 7 H 1 i5 : clX2=chowClass(J,ring(clX)) o5 = 2H 1 ZZ[H ] 1 o5 : ------ 7 H 1 i6 : clX==clX2 o6 = true

## Ways to use chowClass :

• "chowClass(Ideal)"
• "chowClass(Ideal,QuotientRing)"

## For the programmer

The object chowClass is .